2.4
c.
209
QUADRATIC FUNCTIONS
The ball will hit the ground when its height s(t) = 0. Therefore, soJve
- 16t2 + 48t + 8 = 0 for t.
- 16t2
+ 48t + 8 = 0
- 2t + 6t + 1 = 0
r
2
t
Quadratic Formula
See page 101.
• Divide each side by 8.
-(6) ± v'6
2
4(-2)(1)
-
= ---------
• Use the quadratic formula.
2(-2)
-6 ±
v'44
-3 ±
-4
VTT
-2
Using a calculator to approximate the positive root, we find that the ball will hit
the ground in t ;::::; 3.16 seconds. This is also the value of the t-coordinate of
t-intercept in Figure 2.63.
• Try Exercise 70, page 212
Answers to Exercises 9-18 are on pages AA5-AA6.
EXERCISE SET 2.4
e.
In Exercises 1 to 8, match each graph in a. through h.
with the proper quadratic function.
~
1. f(x)
= x2
3
-
3. f(x) = (x - 4) 2
5. f(x) = -2x 2
2. f(x)
d
4. f(x) = (x
b
+2
= x2 + 2
6. f(x)
g
=
+
1
f
3)2
-z-x 2
f.
y
h
+3
X
e
-2
7. f(x) = (x
a.
+
1)2
+
3
8. f(x) = -2(x - 2) 2
c
+2
a
g.
2
h.
y
y
4
b. y
Y
X
2
2
4
2
X
X
2
-4
X
-2
-2
2
c.
4
6
X
d.
In Exercises 9 to 18, use the method of completing the
square to find the standard form of the quadratic
function. State the vertex and axis of symmetry of the
graph of the function and then sketch its graph.
9. f(x) = x 2
~•10. f(x) = x 2
+ 4x + 1
11. f(x) = x 2
-
13. f(x) = x 2
+ 3x +
+ 6x- 1
~12. f(x) = x 2 -
8x + 5
lOx+ 3
X
2
15. f(x) = -x 2
-2
~
2
X
Indicates Enhanced WebAssign problems
~ 14. f(x) = x 2
1
+ 4x + 2
17. f(x) = -3x 2
+ 3x +
+ 7x +
16. f(x) = -x 2 - 2x
7
~18. f(x) = -2x 2
-
2
+
4x
5
+
5
r
210
CHAPTER 2
FUNCTIONS AND GRAPHS
In Exercises 19 to 28, use the vertex formula to determine
the vertex of the graph of the function and write the
function in standard form.
19. f(x) = x 2
-
!Ox
-
10
-25 ),
\'erLe x:
- - "' (,:<. -
fl'21. f(x) = x 2
- 25
Vertex: (3, --9),
22. f(x) = x 2
4
-
~24. f(x) = -x 2
'25. f(x) = 2x
2
3x
-
+7
.,26. f(x) = 3x
Answer or; page AA6.
--:- /- -
+ 4x +
\/ertex: (2, 5)
2
E
J
5
"'-E"
:;;:
Answer on page AA6
27. f(x) = -4x 2 + x + 1
28. f(x)
Answer on page AA6.
=
-5x 2
2
-
1
;
Microgravity 1
ends here
:
8000
6x + 3
-
1
;begins here
8500
10
0
Time (in seconds)
Answer on page AA6.
29. Find the range of f(x) = x 2 - 2x - 1. Determine the values
ofx in the domain ofjfor whichf(x) = 2.
{y: y
/I,Microgravity/~
" 9000
E
1
!Ox+ 2
-
\....-.
1: 9500
'/ertex: <O, -4),
23. f(x) = -x 2 + 6x + 1
and the astronauts experience microgravity. The altitude A(t), in
meters, of the plane t seconds after power was reduced can be
approximated by A(t) = -4.9! 2 + 90t + 9000. The graph is
shown below.
20
If the pilot increases power when the plane descends to
9000 meters, ending microgravity, find the time the astronauts
experience microgravity during one of these maneuvers.
Round to the nearest tenth of a second. 1 8.4 s
2), -1 and 3
~ 30. Find the range off(x) = -x 2
- 6x - 2. Determine the values
of x in the domain ofjfor whichf(x) = 3.
{yjy s 7), -5 and -1
31. Find the range off(x) = - 2x 2 + 5x - l. Determine the values
~ • 48. Soccer Ball Kick The height h(t), in meters, above the ground
Of X in the domain Ofjfor Whichf(x) = 2. (
17[
d3
of a certain soccer ball kick t seconds after the ball is kicked
l_Y~Y""sf' 1 an 2
can be approximated by h(t) = -4.9t 2 + 12.8t. Determine the
.,•32. Find the range off(x) = 2x 2 + 6x - 5. Determine the values
time for which the ball is in the air. Round to the nearest tenth
ofxinthedomainofjforwhichji(x) = 15.f. _ 19\
d7
of a second. 2.6 s
_
1
c
tY!Y""' -2(' -:>an
-
~ 49.
In Exercises 33 to 36, find the real zeros of f and the
x-intercepts of the graph of f.
33. f(x)
=;o
3
h(x) = - - x2
64
~34. f(x) = -x 2 + 6x + 7
x 2 + 2x- 24
-6 and 4; (-6, 0) and (4, 0)
-1 and 7; (-1, 0) and (7, 0)
35. f(x) = 2x 2 + llx + 12
-4
and-~;
(-4, 0)
2 '
36. f(x) = 2x 2 - 9x + 10
and (- ~, o')
2 and 2_; (2, 0) and
o)
2
2
j
+ Sx
•38. f(x)
-16, minimum
2
~39. f(x) = -x
+ 6x +
2
3x + I
- , minimum
8
2
~40. f(x) = -x
II
-
~44. f(x)
I
2
3
4
2
--X
x :5 24
27ft
b. What is the height of the arch 10 feet tp the right of center?
c. How far from the center is the arch 8 feet tall?
6x
"'20. 1 ft from the center
2
= 3x 2
22
5
- ft
16
+ IOx - 3
-
x - I
-
13
12
' '
, mm1murn
4I
-41, minimum
f(x) = --x + 6x + I7
2
46. j(x) =
-
22, maximum
42. f(x) = 3x 2 +
-11, minimum
~45.
-x 2
:5
a. What is the maximum height of the arch?
9, maximum
11, maximum
~41. f(x) = 2x 2 1+
43. f(x) = 5x
=
+ 27, -24
where lxl is the horizontal distance in feet from the center of
the arch to the ground.
(2..2
In Exercises 37 to 46, find the maximum or minimum
value of the function. State whether this value is a
maximum or a minimum.
37. f(x) = x 2
Height of an Arch The height of an arch is given by
-
2
5
-X
+7
35, maximum
529
75
~- =
4
7--- maxirnum
75'
~•
47. Astronaut Training To prepare astronauts for the ~xperience
of zero gravity (technically, microgravity) in space, NASA uses
a specially designed jet. A pilot accelerates the plane upward to
an altitude of approximately 9000 meters and then reduces
power. During the time of reduced power, the plane is in freefall
50. Geometry The sum of the length l and the width w of a rectangular area is 240 meters.
a. Write was a function of l.
,,,
=
240 - '
b. Write the area A as a function of l.
A
=
2401 - 12
c. Find the dimensions that produce the greatest area.
I
=
120 'rn, w
=
1 20 rn
r
2.4
~
• 51. Rectangular Enclosure A veterinarian uses 600 feet of chainlink fencing to enclose a rectangular region and to subdivide the
region into two smaller rectangular regions by placing a fence
_
parallel to one of the sides, as shown in the figure.
21
600
~55.
E(t) = -279.67t 2
b. Write the total area A as a function of l.
A
=
200! -
3
1 00 ft, I
=
3
150 ft
+
82.86t
where 0 :s t :s 0.3 and tis measured in seconds. According to
this model, what is the maximum energy of the bat? Round to
the nearest tenth of a joule. 6.1 jouies
?_
c. Find the dimensions that produce the greatest enclosed area.
=
211
Sports When a softball player swings a bat, the amount of
energy E(t), in joules, that is transferred to the bat can be
approximated by the function
\!\!=~·--·---·
a. Write the width was a function of the length!.
QUADRATIC FUNCTIONS
t-56. •
Geology In June 2001, Mt. Etna in Sicily, Italy, erupted,
sending volcanic bombs (masses of molten lava ejected
from the volcano) into the air. A model of the height h, in
meters, of a volcanic bomb above the crater of the volcano
t seconds after the eruption is given by h(t) = -9.8t 2 + lOOt.
Find the maximum height of a volcanic bomb above the crater
for this eruption. Round to the nearest meter. 255 m
52. Rectangular Enclosure A farmer uses 1200 feet of fence to
enclose a rectangular region and to subdivide the .region into
three smaller rectangular regions by placing fences parallel to
one of the sides. Find the dimensions that produce the greatest
enclosed area. w = 150 ft, I = 300 ft
53. Temperature Fluctuations The temperature T(t), in degrees
Fahrenheit, during the day can be modeled by the equation
T(t) = -0.7t 2 + 9.4t + 59.3, where tis the number of hours
after 6:00 A.M.
~57.
a. At what time is the temperature a maximum? Round to the
nearest minute. 12:43 P.M.
b. What is the maximum temperature? Round to the nearest
degree. 91°F
54. Larvae Survival Soon after insect larvae are hatched, they
must begin to search for food. The survival rate of the larvae
depends on many factors, but the temperature of the environment is one of the most important. For a certain species of
insect, a model of the number oflarvae, N(T), that survive this
searching period is giveri by N(T) = -0.6T 2 + 32.1 T - 350
where Tis the temperature in degrees Celsius.
a. At what temperature will the maximum number of larvae
survive? Round to the nearest degree. 27"C
Sports For a serve to be legal in tennis, the ball must be at
least 3 feet high when it is 39 feet from the server and it must
land in a spot that is less than 60 feet from the server. Does the
path of a ball given by h(x) = -O.OOU - 0.03x + 8, where
h(x) is the height of the ball (in feet) x feet from the server, satisfy the conditions of a legal serve? Yes
58. Sports A pitcher releases a baseball 6 feet above the ground
at a speed of 132 feet per second (90 miles per hour) toward
home plate, which is 60.5 feet away. The height h(x), in feet, of
the ball x feet from home plate can be approximated by
h(x) = -0.0009x 2 + 6. To be considered a strike, the ball
must cross home plate and be at least 2.5 feet high and less
than 5.4 feet high. Assuming the ball crosses home plate, is this
particular pitch a strike? Yes
~59.
Automotive Engineering The fuel efficiency for a certain
midsize car is given by
E(v) = -0.018v 2
+
1.476v
+ 3.4
b. What is the maximum number of surviving larvae? Round
to the nearest integer. 79 larvae
where E(v) is the fuel efficiency in miles per gallon for a car
traveling v miles per hour.
c. Find the x-intercepts, to the nearest integer, for the graph of
this function. (15, 0) and (38, 0)
a. What speed will yield the maximum fuel efficiency? Round
to the nearest mile per hour. 41 rnph
d. I!'W Write a sentenc~ that describes the meaning of the xintercepts in the context of this problem.
b. What is the maximum fuel efficiency for this car? Round to
the nearest mile per gallon. 34 mpg
No larvae survive when the temperature is 15°C (or lower) or 38°C (or higher).
212
CHAPTER 2
FUNCTIONS AND GRAPHS
60. Sports Some football fields are built in a parabolic mound
shape so that water will drain off the field. A model for the
shape of a certain field is given by
r
h(x) = -0.0002348x 2
+ 0.0375x
where h(x) is the height, in feet, of the field at a distance of
x feet from one sideline. Find the maximum height of the field.
Round to the nearest tenth of a foot. 1.5 ft
• 68. Delivery Cost An air freight company has determined that the
cost, in dollars, of delivering x parcels per flight is
C(x) = 2025
+
?x
The price per parcel, in dollars, the company charges to send
x parcels is
p(x) = 22 - O.Olx
Determin<Y'
a. The revenue function
b. The profit function
R(x)
=
-0.01 x 2 ~ 22x
P(x) = -0.01 x 2
c. The company's maximum profit
T
15x - 2025
$3600
d. The price per parcel that yields the maximum profit $14.50
f
j
e. The minimum number of parcels the air freight company
must ship to break even 150 parcels
I
69. Projectile If the initial velocity of a projectile is 128 feet per
second, then its height h, in feet, is a function of time t, in seconds, given by the equation h(t) = -16t 2 + 128t.
Business In Exercises 61 and 62, determine the number of
units x that produce a maximum revenue, in dollars, for
the given revenue function. Also determine the maximum
revenue.
~61. R(x) = 296x - 0.2x 2
740 units yield a maximum revenue of $109,520.
62. R(x) = 810x - 0.6x 2
675 units yield a maximum revenue of $273,375.
I
l
i
I
xz
,
+ 1.68x - 4000
14 000
11,760 units yield a maximum profit of $5878.40.
Business In Exercises 65 and 66, determine the profit
function for the given revenue function and cost
function. Also determine the break-even point or points.
x(102.50- 0.1x); C(x) = 52.50x + 1840
P(x) = -0.1 x 2 + SOx - 1840, break-even points: x = 40 and x = 460
~66. R(x) = x(210- 0.25x); C(x) = 78x + 6399
P(x) = -0.25x 2 + 132x- 6399, break-even points: x = 54 and x = 474
~67. Tour Cost A charter bus company has determined that the
cost, in dollars, of providing x people with a tour is
~65. R(x) =
+ 2.50x
A full tour consists of 60 people. The ticket price per person is
$15 plus $0.25 for each unsold ticket. Determine
a. The revenue function
b. The profit function
R(x) = -0.25x 2
_P(x)
=
-0.25x
c. The company's maximum profit
2
256ft
c. Find the time t when the projectile hits the ground.
t
=
8s
The height in feet of a projectile with an initial
velocity of 64 feet per second and an initial height of 80 feet is
a function of time t in sec.onds given by
63. P(x) = -0.01x 2 + l.?x - 48
85 units yield a maximum profit of $24.25.
C(x) = 180
b. Find the maximum height of the projectile.
~•70. Projectile
Business In Exercises 63 and 64, determine the number of
units x that produce a maximum profit, in dollars, for the
given profit function. Also determine the maximum profit.
64. P(x) = -
a. Find the time t when the projectile achieves its maximum
height. t = 4 s
+ 30.00x
h(t) = -16t 2
+ 64t + 80
a. Find the maximum height of the projectile.
The vertex (2, 144) gives the maximum height of 144 ft.
b. Find the time t when the projectile achieves its maximum
height. t = 2 s
c. Find the time t when the projectile has a height of 0 feet.
Height of 0 ft at t = 5 s
71. Fire Management The height of a stream of water from the
nozzle of a fire hose can be modeled by
y(x) = -0.014x 2
+
1.19x
+5
where y(x) is the height, in feet, of the stream x feet from the
firefighter. What is the maximum height that the stream of water
from this nozzle can reach? Round to the nearest foot. 30 ft
~ 72. Astronaut Training
A weightless environment can be created
in an airplane by flying in a series of parabolic paths. This is one
method that NASA uses to train astronauts for the experience of
weightlessness. Suppose the height h, in feet, of NASA's airplane is modeled by h(t) = -6.6t 2 + 430t + 28,000, where t
is the number of seconds after the plane enters its parabolic
path. Find the maximum height of the plane to the nearest 1000
feet. 35,000 ft
+ 27.50x- 180
$576.25
d. The number of ticket sales that yields the maximum profit
55 tickets
73. Norman Window A Norman window has the shape of a rectangle surmounted by a semicircle. The exterior perimeter of the
window shown in the figure on page 213 is 48 feet. Find the
2.5
r
PROPERTIES OF GRAPHS
213
74. Golden Gate Bridge The suspension cables of the main span
of the Golden Gate Bridge are in the shape of a parabola. If a
coordinate system is drawn as shown, find the quadratic func6.72 ft
tion that models a suspension cable for the main span of the
bridge. f(/) = 0.000112018x 2 , 6
height h and the radius r that will allow the maximum amount
oflight to enter the window. (Hint: Write the area of the window
. ,
as a quadratic function of the radius r.) 48
r=
~--"'
4-c;;-
6.72 rt, n
=
r '=
500
400
300
200
-2000
1000
-1000
2000
X
Properties of Graphs
Symmetry
Even and Odd Functions
Translations of Graphs
PREPARE FOR THIS SECTION
Prepare for this section by completing the following exercises. The answers can be found
on page A12.
Reflections of Graphs
Compressing and Stretching
of Graphs
PSl. For the graph of the parabola whose equation isf(x)
equation of the axis of symmetry? [2.4]
PS2. Forf(x)
=
6, what is the
-2
3x4
= x 2 + , show thatf(-3) = /(3). [2.2]
1
PS3. Forf(x) = 2x 3
PS4. Letf(x)
x
= x 2 + 4x-
-
5x, show thatf(-2) = -/(2). [2.2]
= x 2 and g(x) = x +
3. Findf(a) - g(a) for a
=
-2, -1, 0, 1, 2. [2.2]
3, -1, -3, -3,-1
PSS. What is the midpoint of the line segment betweenP( -a, b) and Q(a, b)? [2.1] (O,
PS6. What is the midpoint of the line segment between P( -a, -b) and Q(a, b)? [2.1]
b)
(O, O)
• Symmetry
Figure 2.64
The graph in Figure 2.64 is symmetric with respect to the line 1. Note that the graph has
the property that if the paper is folded along the dotted line 1 the point A' will coincide
with the point A, the point B' will coincide with the point B, and the point C' will coincide
with the point C. One part of the graph is a mirror image of the rest of the graph across
the line 1.
A graph is symmetric with respect to they-axis if whenever the point given by (x,y)
is on the graph then ( -x,y) is also on the graph. The graph in Figure 2.65 on page 214 is
symmetric with respect to they-axis. A graph is symmetric with respect to the x-axis if